Request for Guidance on Early Project Release

My manager has agreed to release me, but he hasn't mentioned a specific release date. I have seen a few of my seniors who, after getting their manager’s approval for release, were only released after 5–6 months (though I don't know the exact reason for such delays). I believe the same thing might happen in my case. If I get an early release, I can join another project that I am interested in.

Could you please advise me on how to get an early release from the project?